However, I don't want to distribute the Qt libraries along with my programme. I would prefer to give the final user some instructions like:

sudo apt-get install libqt5
sudo apt-get install ...
...

But I don't know what I need to give to the final user. My applications depends on Qt core, gui, serialport, svg, webkitwidgets and network. Where can I find some documentation about this?

I believe that this approach is better than distributing the Qt libraries and running the application through a .sh script because the Qt libraries would be installed in the default location and could be used by more applications. Am I right?

Looks like you are targeting ubuntu, right ? In that case you should rather take a look at how to create a package for them. A properly prepared package will allow your user to just install your software and it will automatically pull in all needed dependencies.